Question

Which substitution is appropriate for \[\sqrt{\frac{x}{x-a}}\], \[\sqrt{\frac{x-a}{x}}\], \[\sqrt{x(x-\mathrm{a})}\], or \[\frac{1}{\sqrt{x(x-\mathrm{a})}}\]?

Options

• \[x=a\tan\theta\]

• \[x=a\sec^2\theta\]

• \[x=a\sin^2\theta\]

• \[x=a\tan^2\theta\]

Answer 1

The forms involving \[x(x-\mathrm{a})\] use \[x=a\sec^2\theta\]. This changes \[x-a\] into an expression containing \[\tan^2\theta\].